Let $C$ be a small category. Isbell duality provides an adjunction $\widehat{C} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}}\widehat{C^{\mathrm{op}}}^{\mathrm{op}}$. If I understand correctly, this notation suggests that $\widehat{C}$ consists of "geometric" objects, whereas $\widehat{C^{\mathrm{op}}}$ consists of "algebraic" objects. Then $\mathcal{O}$ associates to some geometric object $X$ the algebraic object of all global functions on $X$, whereas $\mathrm{Spec}$ associates to some algebraic object $A$ the "affine" geometric object associated to $A$. Note both the unit $\eta_X : X \to \mathrm{Spec}(\mathcal{O}(X))$ and the counit $\varepsilon_A : A \to \mathcal{O}(\mathrm{Spec}(A))$ of this adjunction are given by evaluation. As with every adjunction, we get an equivalence of categories between its fixed points, i.e. those $X$ such that $\eta_X$ is an iso, and those $A$ such that $\varepsilon_A$ is an iso (Isbell-dual objects).

In general this formulation does not make sense, but it is well-known in the following special cases which are also alluded in the nlab article.

1) Algebraic geometry: There is an adjunction $\mathrm{Sch} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}} \mathrm{Ring}^{\mathrm{op}}$. Unit and counit are just evaluation. It restricts to an antiequivalence of categories between affine schemes and rings.

2) Functional analysis: There is an adjunction $\mathrm{Top} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}} {C^*\mathrm{Alg}_1}^{\mathrm{op}}$. Unit and counit are again just evaluation. It restricts to an antiequivalence of categories between compact Hausdorff spaces and commutative unital $C^*$-algebras.

3) Pointless topology: There is an adjunction $\mathrm{Top} {{\mathcal{\Omega} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}} \mathrm{Frm}^{\mathrm{op}} = \mathrm{Loc}$, where $\Omega$ associates to a topological space the frame of its open subsets, and $\mathrm{Spec}$ associates to every locale the space of principal prime ideals. It restricts to an equivalence between sober spaces and spatial locales and is related to Stone duality. This is very, very similar to 2), we just replace $\mathbb{C}$ with the partial order $2$.

Question. Are these adjunctions really special cases of Isbell duality? If not, how are they related? Is there any more general pattern?

I'm also interested in the case of $(2,1)$-categories $C$. Here $\widehat{C}$ should be the category of pseudo-functors $C^{\mathrm{op}} \to \mathrm{Gpd}$. In this setting there is an adjunction similar to 1) between stacks and cocomplete tensor categories, where $\mathcal{O} = \mathrm{Qcoh}$ associates to every stack $X$ the category of quasi-coherent modules, which may be imagined as categorified global functions on $X$. The fixed points are the tensorial stacks which I study currently. Where does this adjunction arise in the literature? It is similar to an adjunction from derived algebraic geometry (Ben-Zvi, Nadler, Prop. 3.1).

There are related conversations between Jim Dolan and Todd Trimble, which already answer my questions partially.

1 Answer
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This seems to be an informal answer: Whenever we have an algebraic object $\mathbb{A}$ of type $A$ in a geometric category $G$, we get an adjunction: $\mathrm{O} := \mathrm{Hom}(-,\mathbb{A}) : G \to A^{op}$ is left adjoint to $\mathrm{Spec} := \mathrm{Hom}(-,\mathbb{A}): A^{op} \to G$ where the latter hom-set is endowed with some kind of Zariski topology. The four adjunctions from above are special cases:

1) Consider the ring object $\mathbb{A}^1 := \mathrm{Spec}(\mathbb{Z}[x])$ in the category of schemes.

You might want to be a little bit more precise about what you're saying. The reason why $\textrm{Hom}(-, \mathbb{A}^1)$ lands in the category of rings is because the theory of rings is an algebraic theory, and hence a finite limit theory a fortiori; but of course, the Yoneda embedding is left exact, so this means that each $\textrm{Hom}(S, \mathbb{A}^1)$ is a ring object in $\textbf{Set}$ in a natural way. Unfortunately this reasoning doesn't work for frames, because the theory of frames is not a finite limit theory. These objects are known as ‘dualising objects’ and some other names.
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Zhen LinMar 8 '12 at 17:39

I've read that objects which belong to two concrete categories at once are called schizophrenic. Usually schizophrenic objects induce adjunctions.
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Martin BrandenburgOct 14 '14 at 7:51